I will give you some intuitive remarks for some inspiration on the proofs.
For the first one, notice that if m divides n then n = pm where p is a integer.
Since n and m are both natural numbers p then must be a natural number as well.
Now we know that basically we want to prove that if a is congruent to b mod n then a is congruent to b mod "a factor of n" (this is cause n = pm).
Tell me if you need more clarification.
For the second proof, I would just draw a Venn diagram and prove that the two intersections cover identical regions.